thomas teubner - standard model (2008)

8/6/2019 Thomas Teubner - Standard Model (2008)

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THE STANDARD MODEL

Thomas Teubner

Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, U.K.

Lectures presented at the School for Young High Energy Physicists,

Somerville College, Oxford, September 2008.


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Contents

1 QED as an Abelian Gauge Theory 7

1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Covariant Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Non-Abelian Gauge Theories 15

2.1 Global Non-Abelian Transformations . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Non-Abelian Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 The Lagrangian for a General Non-Abelian Gauge Theory . . . . . . . . . . 20

2.5 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Quantum Chromodynamics 25

3.1 Running Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Quark (and Gluon) Confinement . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 -Parameter of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Spontaneous Symmetry Breaking 32

4.1 Massive Gauge Bosons and Renormalizability . . . . . . . . . . . . . . . . . 32

4.2 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 The Abelian Higgs Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4 Goldstone Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37


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4.5 The Unitary Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.6 R Gauges (Feynman Gauge) . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 The Standard Model with one Family 43

5.1 Left- and Right- Handed Fermions . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Symmetries and Particle Content . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Kinetic Terms for the Gauge Bosons . . . . . . . . . . . . . . . . . . . . . . 46

5.4 Fermion Masses and Yukawa Couplings . . . . . . . . . . . . . . . . . . . . 47

5.5 Kinetic Terms for Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.6 The Higgs Part and Gauge Boson Masses . . . . . . . . . . . . . . . . . . . . 52

5.7 Classifying the Free Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 Additional Generations 63

6.1 A Second Quark Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2 Flavour Changing Neutral Currents . . . . . . . . . . . . . . . . . . . . . . . 65

6.3 Adding Another Lepton Generation . . . . . . . . . . . . . . . . . . . . . . 66

6.4 Adding a Third Generation (of Quarks) . . . . . . . . . . . . . . . . . . . . 68

6.5 CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7 Neutrinos 75

7.1 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.2 Oscillations in Quantum Mechanics (in Vaccum and Matter) . . . . . . . . . 78

7.3 The See-Saw Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

8 Supersymmetry 85


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8.1 Why Supersymmetry? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.2 A New Symmetry: Boson Fermion . . . . . . . . . . . . . . . . . . . . . 868.3 The Supersymmetric Harmonic Oscillator . . . . . . . . . . . . . . . . . . . 88

8.4 Supercharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

8.5 Superfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8.6 The MSSM Particle Content (Partially) . . . . . . . . . . . . . . . . . . . . . 93

8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


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Introduction

An important feature of the Standard Model (SM) is that it works: it is consistent with,

or verified by, all available data, with no compelling evidence for physics beyond.1 Secondly,

it is a unified description, in terms of gauge theories of all the interactions of knownparticles (except gravity). A gauge theory is one that possesses invariance under a set of

local transformations, i.e. transformations whose parameters are space-time dependent.

Electromagnetism is a well-known example of a gauge theory. In this case the gauge trans-

formations are local complex phase transformations of the fields of charged particles, and

gauge invariance necessitates the introduction of a massless vector (spin-1) particle, called

the photon, whose exchange mediates the electromagnetic interactions.

In the 1950s Yang and Mills considered (as a purely mathematical exercise) extending gauge

invariance to include local non-abelian (i.e. non-commuting) transformations such as SU(2).

In this case one needs a set of massless vector fields (three in the case of SU(2)), which were

formally called Yang-Mills fields, but are now known as gauge fields.

In order to apply such a gauge theory to weak interactions, one considers particles which

transform into each other under the weak interaction, such as a u-quark and a d-quark, or

an electron and a neutrino, to be arranged in doublets of weak isospin. The three gauge

bosons are interpreted as the W and Z bosons, that mediate weak interactions in the same

way that the photon mediates electromagnetic interactions.

The difficulty in the case of weak interactions was that they are known to be short range, me-

diated by very massive vector bosons, whereas Yang-Mills fields are required to be massless

in order to preserve gauge invariance. The apparent paradox was solved by the applica-

tion of the Higgs mechanism. This is a prescription for breaking the gauge symmetry

spontaneously. In this scenario one starts with a theory that possesses the required gauge

invariance, but where the ground state of the theory is notinvariant under the gauge transfor-

mations. The breaking of the invariance arises in the quantization of the theory, whereas the

Lagrangian only contains terms which are invariant. One of the consequences of this is that

the gauge bosons acquire a mass and the theory can thus be applied to weak interactions.

Spontaneous symmetry breaking and the Higgs mechanism have another extremely impor-

tant consequence. It leads to a renormalizable theory with massive vector bosons. This

means that one can carry out a programme of renormalization in which the infinities that

1In saying so we have taken the liberty to allow for neutrino masses (see chapter 7) and discarded some

deviations in electroweak precision measurements which are far from conclusive; however, note that there is

a 3.4 deviation between measurement and SM prediction ofg2 of the muon, see the remarks in chapter 8.


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arise in higher-order calculations can be reabsorbed into the parameters of the Lagrangian

(as in the case of QED). Had one simply broken the gauge invariance explicitly by adding

mass terms for the gauge bosons, the resulting theory would not have been renormalizable

and therefore could not have been used to carry out perturbative calculations. A consequence

of the Higgs mechanism is the existence of a scalar (spin-0) particle, the Higgs boson.

The remaining step was to apply the ideas of gauge theories to the strong interaction. The

gauge theory of the strong interaction is called Quantum Chromo Dynamics (QCD). In this

theory the quarks possess an internal property called colour and the gauge transformations

are local transformations between quarks of different colours. The gauge bosons of QCD are

called gluons and they mediate the strong interaction.

The union of QCD and the electroweak gauge theory, which describes the weak and elec-

tromagnetic interactions, is known as the Standard Model. It has a very simple structure

and the different forces of nature are treated in the same fashion, i.e. as gauge theories.It has eighteen fundamental parameters, most of which are associated with the masses of

the gauge bosons, the quarks and leptons, and the Higgs. Nevertheless these are not all

independent and, for example, the ratio of the W and Z boson masses are (correctly) pre-

dicted by the model. Since the theory is renormalizable, perturbative calculations can be

performed at higher order that predict cross sections and decay rates for both strongly and

weakly interacting processes. These predictions, when confronted with experimental data,

have been confirmed very successfully. As both predictions and data are becoming more and

more precise, the tests of the Standard Model are becoming increasingly stringent.


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1 QED as an Abelian Gauge Theory

The aim of this lecture is to start from a symmetry of the fermion Lagrangian and show

that gauging this symmetry (= making it well behaved) implies classical electromagnetism

with its gauge invariance, the ee interaction, and that the photon must be massless.

1.1 Preliminaries

In the Field Theory lectures at this school, the quantum theory of an interacting scalar

field was introduced, and the voyage from the Lagrangian to the Feynman rules was made.

Fermions can be quantised in a similar way, and the propagators one obtains are the Green

functions for the Dirac wave equation (the inverse of the Dirac operator) of the QED/QCD

course. In this course, I will start from the Lagrangian (as opposed to the wave equation) ofa free Dirac fermion, and add interactions, to construct the Standard Model Lagrangian in

classical field theory. That is, the fields are treated as functions, and I will not discuss creation

and annihiliation operators. However, to extract Feynman rules from the Lagrangian, I will

implicitly rely on the rules developed for scalar fields in the Field Theory course.

1.2 Gauge Transformations

Consider the Lagrangian density for a free Dirac field :

L = (i m) (1.1)

This Lagrangian density is invariant under a phase transformation of the fermion field

eiQ, eiQ, (1.2)

where Q is the charge operator (Q = +, Q = ), is a real constant (i.e. independentof x) and is the conjugate field.

The set of all numbers ei form a group2. This particular group is abelian which is to

say that any two elements of the group commute. This just means that

ei1ei2 = ei2ei1 . (1.3)

2A group is a mathematical term for a set, where multiplication of elements is defined and results in

another element of the set. Furthermore, there has to be a 1 element (s.t. 1 a = a) and an inverse (s.t.a a1 = 1) for each element a of the set.


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This particular group is called U(1) which means the group of all unitary 1 1 matrices. Aunitary matrix satisfies U+ = U1 with U+ being the adjoint matrix.

We can now state the invariance of the Lagrangian eq. (1.1) under phase transformations in a

more fancy way by saying that the Lagrangian is invariant under global U(1) transformations.

By global we mean that does not depend on x.

For the purposes of these lectures it will usually be sufficient to consider infinitesimal group

transformations, i.e. we assume that the parameter is sufficiently small that we can expand

in and neglect all but the linear term. Thus we write

ei = 1 i + O(2). (1.4)

Under such infinitesimal phase transformations the field changes according to

+ = + iQ , (1.5)and the conjugate field by

+ = + i Q = i , (1.6)

such that the Lagrangian density remains unchanged (to order ).

At this point we should note that global transformations are not very attractive from a

theoretical point of view. The reason is that making the same transformation at every

space-time point requires that all these points know about the transformation. But if Iwere to make a certain transformation at the top of Mont Blanc, how can a point somewhere

in England know about it? It would take some time for a signal to travel from the Alps to

England.

Thus, we have two options at this point. Either, we simply note the invariance of eq. (1.1)

under global U(1) transformations and put this aside as a curiosity, or we insist that in-

variance under gauge transformations is a fundamental property of nature. If we take the

latter option we have to require invariance under local transformations. Local means that

the parameter of the transformation, , now depends on the space-time point x. Such local

(i.e. space-time dependent) transformations are called gauge transformations.

If the parameter depends on the space-time point then the field transforms as follows

under infinitesimal transformations

(x) = i (x) (x); (x) = i (x) (x). (1.7)

Note that the Lagrangian density eq. (1.1) now is no longer invariant under these trans-

formations, because of the partial derivative between and . This derivative will act on


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the space-time dependent parameter (x) such that the Lagrangian density changes by an

amount L, whereL = (x) [Q(x)] (x). (1.8)

The square brackets in [Q(x)] are introduced to indicate that the derivative acts only

inside the brackets. It turns out that we can restore gauge invariance if we assume that thefermion field interacts with a vector field A, called a gauge field, with an interaction term

e AQ (1.9)added to the Lagrangian density which now becomes

L = (i ( + i e Q A) m) . (1.10)In order for this to work we must also assume that apart from the fermion field transform-

ing under a gauge transformation according to eq. (1.7) the gauge field, A, also changes

according to

eQA eQ(A + A(x)) = eQ A + Q (x). (1.11)So A(x) = Q (x)/e.

Exercise 1.1

Using eqs. (1.7) and (1.11) show that under a gauge transformation

(e A) = (x) [Q(x)] (x).

This change exactly cancels with eq. (1.8), so that once this interaction term has been added

the gauge invariance is restored. We recognize eq. (1.10) as being the fermionic part of the

Lagrangian density for QED, where e is the electric charge of the fermion and A is the

photon field.

In order to have a proper quantum field theory, in which we can expand the photon field A

in terms of creation and annihilation operators for photons, we need a kinetic term for the

photon, i.e. a term which is quadratic in the derivative of the field A. Without such a term

the Euler-Lagrange equation for the gauge field would be an algebraic equation and we could

use it to eliminate the gauge field altogether from the Lagrangian. We need to ensure that

in introducing a kinetic term we do not spoil the invariance under gauge transformations.

This is achieved by defining the field strength tensor, F, as

F A A, (1.12)where the derivative is understood to act on the A-field only.3 It is easy to see that under

the gauge transformation eq. (1.11) each of the two terms on the right hand side of eq. (1.12)

3Strictly speaking we should therefore write F = [A ] [A]; you will find that the brackets areoften omitted.


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change, but the changes cancel out. Thus we may add to the Lagrangian any term which

depends on F (and which is Lorentz invariant, thus, with all Lorentz indices contracted).

Such a term is aFF. This gives the desired term which is quadratic in the derivative of

the field A. If we choose the constant a to be 1/4 then the Lagrange equations of motionmatch exactly (the relativistic formulation of) Maxwells equations.

4

We have thus arrived at the Lagrangian density for QED, but from the viewpoint of de-

manding invariance under U(1) gauge transformations rather than starting with Maxwells

equations and formulating the equivalent quantum field theory.

The Lagrangian density for QED is:

L = 14

FF+ (i ( + i e Q A) m) . (1.13)

Exercise 1.2Starting with the Lagrangian density for QED write down the Euler-Lagrange

equations for the gauge field A and show that this results in Maxwells equa-

tions.

In the Field Theory lectures, we have seen that a term 4 in the Lagrangian gave 4! as

the coupling of four s in perturbation theory. Neglecting the combinatoric factors, it is

plausible that eq. (1.13) gives the ee Feynman Rule used in the QED course, ie, fornegatively charged particles.

Note that we are not allowed to add a mass term for the photon. A term such as M2AA

added to the Lagrangian density is not invariant under gauge transformations as it would

lead to

L = 2M2

eA(x)(x) = 0. (1.14)

Thus the masslessness of the photon can be understood in terms of the requirement that the

Lagrangian be gauge invariant.

1.3 Covariant Derivatives

Before leaving the abelian case, it is useful to introduce the concept of a covariant deriva-

tive. This is not essential for abelian gauge theories, but will be an invaluable tool when

we extend these ideas to non-abelian gauge theories.

4The determination of this constant a is the only place that a match to QED has been used. The rest

of the Lagrangian density is obtained purely from the requirement of local U(1) invariance. A different

constant would simply mean a different normalization of the photon field.


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The covariant derivative D is defined to be

D + i e A. (1.15)

It has the property that given the transformations of the fermion field eq. (1.7) and the gauge

field eq. (1.11) the quantity D transforms in the same way under gauge transformations

as .

Exercise 1.3

Show that under an infinitesimal gauge transformation D transforms as

D D + (D) with (D) = i(x)D.

We may thus rewrite the QED Lagrangian density as

L = 14

FF+ (iD m) . (1.16)

Furthermore the field strength F can be expressed in terms of the commutator of two

covariant derivatives, i.e.

F = ie

[D, D] = ie

[, ] + [, A] + [A, ] + i e [A, A]

= A A, (1.17)

where in the last line we have adopted the conventional notation again and left out the

square brackets. Notice that when using eq. (1.17) the derivatives act only on the A-field.

1.4 Gauge Fixing

The guiding principle of this chapter has been to hold onto the U(1) symmetry. This forced

us to introduce a new massless field A which we could interpret as the photon. In this

subsection we will try to quantise the photon field (e.g. calculate its propagator) by naively

following the prescription used for scalars and fermions, which will not work. This should not

be surprising, because A has four real components, introduced to maintain gauge symmetry.

However the physical photon has two polarisation states. This difficulty can be resolved by

fixing the gauge (breaking our precious gauge symmetry) in the Lagrangian in such a way

as to maintain the gauge symmetry in observables.5

5The gauge symmetry is also preserved in the Path Integral, which is a sum over all field configurations

weighted by exp{i Ld4x}. In path integral quantisation, which is an alternative to the canonical approachused in the Field Theory lectures, Green functions are calculated from the path integral and it is unimportant

that the gauge symmetry seems broken in the Lagrangian.


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In general, if the part of the action that is quadratic in some field (x) is given in terms of

the Fourier transform (p) by

S =

d4p (p)O(p)(p), (1.18)

then the propagator for the field may be written as

i O1(p). (1.19)

In the case of QED the part of the Lagrangian that is quadratic in the photon field is given

by 1/4 FF = 1/2 A (g + ) A, where we have used partial integrationto obtain the second expression. In momentum space, the quadratic part of the action is

then given by

SA =

d4p1

2A(p)

gp2 + pp

A(p). (1.20)

Unfortunately the operator (gp2 + pp) does not have an inverse. This can be mosteasily seen by noting (gp2 + pp)p = 0. This means that the operator (gp2 + pp)has an eigenvector (p) with eigenvalue 0 and is therefore not invertible. Thus it seems we

are not able to write down the propagator of the photon. We solve this problem by adding

to the Lagrangian density a gauge fixing term

12(1 ) (A

)2 . (1.21)

With this term included (again in momentum space), SA becomes

SA =

d4p1

2A(p)

gp2

1 pp

A(p), (1.22)

and, noting the relationgp

2 +

1 pp

g pp

p2

= p2g , (1.23)

we see that the propagator for the photon may now be written as

i g

pp

p2 1

p2. (1.24)

The special choice = 0 is known as the Feynman gauge. In this gauge the propagator

eq. (1.24) is particularly simple and we will use it most of the time.

This procedure of gauge fixing seems strange: first we worked hard to get a gauge invariant

Lagrangian, and then we spoil gauge invariance by introducing a gauge fixing term.

The point is that we have to fix the gauge in order to be able to perform a calculation.

Once we have computed a physical quantity, the dependence on the gauge cancels. In other


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words, it does not matter how we fix the gauge, and in particular, what value for we take.

The choice = 0 is simply a matter of convenience. A more careful procedure would be to

leave arbitrary and check that all -dependence in the final result cancels. This gives us

a strong check on the calculation, however, at the price of making the computation much

more tedious.

The procedure of fixing the gauge in order to be able to perform a calculation, even though

the final result does not depend on how we have fixed the gauge, can be understood by

the following analogy. Assume we wanted to calculate some scalar quantity (say the time

it takes for a point mass to get from one point to another) in our ordinary 3-dimensional

Euclidean space. To do so, we choose a coordinate system, perform the calculation and

get our final result. Of course, the result does not depend on how we choose the coordinate

system, but in order to be able to perform the calculation we have to fix it somehow. Picking

a coordinate system corresponds to fixing a gauge, and the independence of the result on thecoordinate system chosen corresponds to the gauge invariance of physical quantities. To take

this one step further we remark that not all quantities are independent of the coordinate

system. For example, the x-coordinate of the position of the point mass at a certain time

depends on our choice. Similarly, there are important quantities that are gauge dependent.

One example is the gauge boson propagator given in eq. (1.24). However, all measurable

quantities (observables) are gauge invariant. This is where our analogy breaks down: in

our Euclidean example there are measurable quantities that do depend on the choice of the

coordinate system.

Finally we should mention that eq. (1.21) is by far not the only way to fix the gauge but

it will be sufficient for these lectures to consider gauges defined through eq. (1.21). These

gauges are called covariant gauges.

1.5 Summary

It is possible for the Lagrangian for a (complex) Dirac field to be invariant under

local U(1) transformations (phase rotations), in which the phase parameter dependson space-time. In order to accomplish this we include an interaction with a vector

gauge boson which transforms under the local (gauge) transformation according to

eq. (1.11).

This interaction is encoded by replacing the derivative by the covariant derivativeD defined by eq. (1.15). D transforms under gauge transformations as e

i D .

The kinetic term for the gauge boson is 14

FF, where F is proportional to the


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commutator [D, D] and is invariant under gauge transformations.

The gauge boson must be massless, since a term proportional to AA is not invariantunder gauge transformations and hence not included in the Lagrangian.

The resulting Lagrangian is identical to that of QED. In order to define the propagator we have to specify a certain gauge; the resulting

gauge dependence cancels in physical observables.


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2 Non-Abelian Gauge Theories

In this lecture, the gauge concept will be constructed so that the gauge bosons have self-

interactions as are observed among the gluons of QCD, and the W, Z and of the

electroweak sector. However, the gauge bosons will still be massless. (We will see how togive the W and Z their observed masses in the Higgs chapter.)

2.1 Global Non-Abelian Transformations

We apply the ideas of the previous lecture to the case where the transformations do not

commute with each other, i.e. the group is non-abelian.

Consider n free fermion fields

{i

}, arranged in a multiplet :

=

1

2

.

.

n

(2.1)

for which the Lagrangian density is

L= (i

m) ,

i (i m) i, (2.2)

where the index i is summed from 1 to n. Eq. (2.2) is therefore a shorthand for

L = 1 (i m) 1 + 2 (i m) 2 + . . . . (2.3)

The Lagangian density (2.2) is invariant under (space-time independent) complex rotations

in i space:

U,

U, (2.4)

where U is an n n matrix such that

UU = 1, det[U] = 1. (2.5)

The transformation (2.4) is called an internal symmetry, which rotates the fields (e.g. quarks

of different colour) among themselves.

The group of matrices satisfying the conditions (2.5) is called SU(n). This is the group

of special, unitary n n matrices. Special in this context means that the determinant is


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equal to 1. In order to specify an SU(n) matrix completely we need n2 1 real parameters.Indeed, we need 2n2 real parameters to determine an arbitrary complex n n matrix. Butthere are n2 constraints due to the unitary requirements and one additional constraint due

to the requirement det = 1.

An arbitrary SU(n) matrix can be written as

U = ein21

a=1aTa eiaTa (2.6)

where we again have adopted Einsteins summation convention. The a, a {1 . . . n2 1},are real parameters, and the Ta are called the generators of the group.

Exercise 2.1

Show that the unitarity of the SU(n) matrices entails hermiticity of the gen-

erators and that the requirement of det = 1 means that the generators have

to be traceless.

In the case of U(1) there was just one generator. Here we have n2 1 generators Ta.There is still some freedom left of how to normalize the generators. We will adopt the usual

normalization convention

tr(TaTb) =1

2ab. (2.7)

The reason we can always enforce eq. (2.7) is that tr(TaTb) is a real matrix symmetric in

a b. Thus it can be diagonalized. If you have problems getting on friendly terms with theconcept of generators, for the moment you can think of them as traceless, hermitian n nmatrices. (This is, however, not the complete picture.)

The crucial new feature of the group SU(n) is that two elements of SU(n) generally do not

commute, i.e.

eia1Taei

b2Tb = eib2Tbeia1 Ta (2.8)

(compare to eq. (1.3)). To put this in a different way, the group algebra is not trivial. For

the commutator of two generators we have

[Ta, Tb] ifabcTc = 0 (2.9)

where we defined the structure constants of the group, fabc, and used the summation conven-

tion again. The structure constants are totally antisymmetric. This can be seen as follows:

from eq. (2.9) it is obvious that fabc = fbac. To convince us of the antisymmetry in theother indices as well, we note that multiplying eq. (2.9) by Td and taking the trace, using

eq. (2.7), we get 1/2 ifabd = tr(TaTbTd) tr(TbTaTd) = tr(TaTbTd) tr(TaTdTb).


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2.2 Non-Abelian Gauge Fields

Now suppose we allow the transformation U to depend on space-time. Then the Lagrangian

density changes by L under this non-abelian gauge transformation, where

L = U (U) . (2.10)The local gauge symmetry can be restored by introducing a covariant derivative D, giving

interactions with gauge bosons, such that

DU(x)(x) = U(x)D(x). (2.11)

This is like the electromagnetic case, except that D is now a matrix,

iD = iI gA (2.12)

where A = Ta

Aa. It contains n

2

1 vector (spin one) gauge bosons, Aa, one for each

generator of SU(n). Under a gauge transformation U, A should transform as

A UAU + ig

(U) U. (2.13)

This ensures that the Lagrangian density

L = (iD m) (2.14)is invariant under local SU(n) gauge transformations. It can be checked that eq. (2.13)

reduces to the gauge transformation of electromagnetism in the abelian limit.

Exercise 2.2

(For algebraically ambitious people): perform an infinitesimal gauge transfor-

mation on , and D, using (2.6), and show that to linear order in the a,

D is invariant.

Exercise 2.3

Show that in the SU(2) case, the covariant derivative is

iD = i g2 W3 g2 (W1 iW2)

g2

(W1 + iW2) i +

g2

W3

,and find the usual charged current interactions for the lepton doublet

=

e

by defining W = (W1 iW2)/2.


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Exercise 2.4

Include the U(1) hypercharge interaction in the previous question; show that

the covariant derivative acting on the lepton doublet (of hypercharge Y =

1/2) is

iD = i g2 W3 gY B g2 (W1 iW2)

g2

(W1 + iW2) i +

g2

W3 gY B .

Define ZA

= cos W sin W

sin W cos W

W3B

and write the diagonal (neutral) interactions in terms of Z and A. Extract

sin W in terms ofg and g. (Recall that the photon does not interact with the

neutrino.)

The kinetic term for the gauge bosons is again constructed from the field strengths Fa which

are defined from the commutator of two covariant derivatives,

F = ig

[D, D] , (2.15)

where the matrix F is given by

F = TaFa, (2.16)

withFa = A

a Aa g fabc AbAc. (2.17)

Notice that F is gauge variant, unlike the U(1) case. We know the transformation of D

from (2.13), so

[D, D] U [D, D] U. (2.18)The gauge invariant kinetic term for the gauge bosons is therefore

12

Tr FF = 1

4FaF

a, (2.19)

where the trace is in SU(n) space, and summation over the index a is implied.

In sharp contrast with the abelian case, this term does not only contain terms which are

quadratic in the derivatives of the gauge boson fields, but also the terms

g fabc(Aa)A

bA

c

1

4g2fabcfadeAbA

cA

dA

e. (2.20)

This means that there is a very important difference between abelian and non-abelian gauge

theories. For non-abelian gauge theories the gauge bosons interact with each other via both


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three-point and four-point interaction terms. The three point interaction term contains

a derivative, which means that the Feynman rule for the three-point vertex involves the

momenta of the particles going into the vertex. We shall write down the Feynman rules in

detail later.

Once again, a mass term for the gauge bosons is forbidden, since a term proportional to

AaAa is not invariant under gauge transformations.

2.3 Gauge Fixing

As in the case of QED, we need to add a gauge-fixing term in order to be able to derive a

propagator for the gauge bosons. In Feynman gauge this means adding the term 12

(Aa)2

to the Lagrangian density, and the propagator (in momentum space) becomes

i ab gp2

.

There is one unfortunate complication, which is mentioned briefly here for the sake of com-

pleteness, although one only needs to know about it for the purpose of performing higher

loop calculations with non-abelian gauge theories:

If one goes through the formalism of gauge-fixing carefully, it turns out that at higher

orders extra loop diagrams emerge. These diagrams involve additional particles that are

mathematically equivalent to interacting scalar particles and are known as a Faddeev-Popovghosts. For each gauge field there is such a ghost field. These are not to be interpreted

as physical scalar particles which could in principle be observed experimentally, but merely

as part of the gauge-fixing programme. For this reason they are referred to as ghosts.

Furthermore they have two peculiarities:

1. They only occur inside loops. This is because they are not really particles and cannot

occur in initial or final states, but are introduced to clean up a difficulty that arises in

the gauge-fixing mechanism.

2. They behave like fermions even though they are scalars (spin zero). This means that

we need to count a minus sign for each loop of Faddeev-Popov ghosts in any Feynman

diagram.

We shall display the Feynman rules for these ghosts later.

Thus, for example, the Feynman diagrams which contribute to the one-loop corrections to

the gauge boson propagator are


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+ - -

(a) (b) (c) (d)

Diagram (a) involves the three-point interaction between the gauge bosons, diagram (b)

involves the four-point interaction between the gauge bosons, diagram (c) involves a loop of

fermions, and diagram (d) is the extra diagram involving the Faddeev-Popov ghosts. Note

that both diagrams (c) and (d) have a minus sign in front of them because both fermions

and Faddeev-Popov ghosts obey Fermi statistics.

2.4 The Lagrangian for a General Non-Abelian Gauge Theory

Let us summarize what we have found so far: Consider a gauge group G of dimensionN (for SU(n) : N n2 1), whose N generators, Ta, obey the commutation relationsTa, Tb

= ifabcT

c, where fabc are called the structure constants of the group.

The Lagrangian density for a gauge theory with this group, with a fermion multiplet i, is

given (in Feynman gauge) by

L = 14

FaFa+ i (D mI) 1

2(Aa)

2 + LFP (2.21)

where

Fa = Aa Aa g fabcAbAc, (2.22)

D = I + i g TaAa (2.23)

and

LFP =

a

a + g facbaAc

(b). (2.24)

Under an infinitesimal gauge transformation the N gauge bosons Aa change by an amount

that contains a term which is not linear in Aa:

Aa(x) = fabcAb(x)c(x) +1

g

a(x), (2.25)

whereas the field strengths Fa transform by a change

Fa(x) = fabc Fb(x) c. (2.26)


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In other words, they transform as the adjoint representation of the group (which has as

many components as there are generators). This means that the quantity FaFa (summa-

tion over a,, implied) is invariant under gauge transformations.

2.5 Feynman Rules

The Feynman rules for such a gauge theory can be read off directly from the Lagrangian. As

mentioned previously, the propagators are obtained by taking all terms bilinear in the field

and inverting the corresponding operator (and multiplying by i). The rules for the vertices

are obtained by simply taking (i times) the factor which multiplies the corresponding term

in the Lagrangian. The explicit rules are given in the following.

Vertices:

(Note that all momenta are defined as flowing into the vertex!)

a

p1

c

p3

b

p2g fabc

g(p1 p2) + g (p2 p3) + g (p3 p1)

i g2feabfecd (gg gg)i g2feacfebd (gg gg)i g2feadfebc (gg gg)

a b

d c

a

j i

i g (Ta)ij

a

c bq

g fabc q


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Propagators:

Gluon: i ab g/p2pa

b

Fermion: i ij(p + m)/(p

2 m2)pi j

Faddeev-Popov ghost: i ab/p2pa b

2.6 An Example

As an example of the application of these Feynman rules, we consider the process of Compton

scattering, but this time for the scattering of non-abelian gauge bosons and fermions, rather

than photons. We need to calculate the amplitude for a gauge boson of momentum p2 and

gauge label a to scatter off a fermion of momentum p1 and gauge label i producing a fermion

of momentum p3 and gauge label j and a gauge boson of momentum p4 and gauge label

b. Note that i, j {1 . . . n} whereas a, b {1 . . . n2 1}. In addition to the two Feynmandiagrams one gets in the QED case there is a third diagram involving the self-interaction of

the gauge bosons.

i k jp1 p3

(p1 + p2)

p2 p4

a b

i k jp1 p3

p2 p4

a b

i jp1 p3

(p4 p2) c

p2 p4a b

(a) (b) (c)

We will assume that the fermions are massless (i.e. that we are at sufficiently high energies

so that we may neglect their masses), and work in terms of the Mandelstam variables

s = (p1 + p2)2 = (p3 + p4)

2,

t = (p1 p3)2 = (p2 p4)2,u = (p1 p4)2 = (p2 p3)2.

The polarizations are accounted for by contracting the amplitude obtained for the above

diagrams with the polarization vectors (2) and (4). Each diagram consists of two

vertices and a propagator and so their contributions can be read off from the Feynman rules.


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For diagram (a) we get

(2)(4)uj(p3)

i g (Tb)kj

i

(p1 + p2)s

i g (Ta)ik

ui(p1)

= i g2s

(2)(4)u (p3) ( (p1 + p2)) TbTau(p1).

For diagram (b) we get

(2)(4)uj(p3)

i g (Ta)kj

i

(p1 p4)u

i g (Tb)ik

ui(p1)

= i g2

u(2)(4)u (p3) (

(p1 p4))

TaTb

u(p1).

Note that here the order of the T matrices is the other way around compared to diagram(a).

Diagram (c) involves the three-point gauge-boson self-coupling. Since the Feynman rule for

this vertex is given with incoming momenta, it is useful to replace the outgoing gauge-boson

momentum p4 by p4 and understand this to be an incoming momentum. Note that theinternal gauge-boson line carries momentum p4 p2 coming into the vertex. The threeincoming momenta that are to be substituted into the Feynman rule for the vertex are

therefore p2, p4, p4 p2. The vertex thus becomes

g fabc (g(p2 + p4) + g(p2 2p4) + g(p4 2p2)) ,

and the diagram gives

(2)(4)u

j(p3)i g (Tc)ij

ui(p1)

ig

t

(g fabc) (g(p2 + p4) + g(p2 2p4) + g(p4 2p2))

=

i

g2

t(2)

(4)u (p3) Ta, Tb

u(p1)

g(p2 + p4)

2(p4)g

2(p2)g

,

where in the last step we have used the commutation relation eq. (2.9) and the fact that the

polarization vectors are transverse so that p2 (2) = 0 and p4 (4) = 0.

Exercise 2.4

Draw all the Feynman diagrams for the tree level amplitude for two gauge

bosons with momenta p1 and p2 to scatter into two gauge bosons with momenta

q1 and q2. Label the momenta of the external gauge boson lines.


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2.7 Summary

A non-abelian gauge theory is one in which the Lagrangian is invariant under localtransformations of a non-abelian group.

This invariance is achieved by introducing a gauge boson for each generator of thegroup. The partial derivative in the Lagrangian for the fermion field is replaced by a

covariant derivative as defined in eq. (2.23).

The gauge bosons transform under infinitesimal gauge transformations in a non-linearway given by eq. (2.25).

The field strengths, Fa, are obtained from the commutator of two covariant derivativesand are given by eq. (2.22). They transform as the adjoint representation under gauge

transformations such that the quantity Fa

Fa

is invariant.

FaFa contains terms which are cubic and quartic in the gauge bosons, indicatingthat these gauge bosons interact with each other.

The gauge-fixing mechanism leads to the introduction of Faddeev-Popov ghosts whichare scalar particles that occur only inside loops and obey Fermi statistics.


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3 Quantum Chromodynamics

Quantum Chromodynamics (QCD) is the theory of the strong interaction. It is nothing but

a non-abelian gauge theory with the group SU(3). Thus, the quarks are described by a

field i where i runs from 1 to 3. The quantum number associated with the label i is calledcolour. The eight gauge bosons which have to be introduced in order to preserve local gauge

invariance are the eight gluons. These are taken to be the carriers which mediate the strong

interaction in the same way that photons are the carriers which mediate the electromagnetic

interactions.

The Feynman rules for QCD are therefore simply the Feynman rules listed in the previous

lecture, with the gauge coupling constant, g, taken to be the strong coupling, gs, (more

about this later), the generators Ta taken to be the eight generators of SU(3) in the triplet

representation, and fabc

, a, b, c, = 1 . . . 8 are the structure constants ofSU(3) (you can lookthem up in a book but normally you will not need their explicit form).

Thus we now have a quantum field theory which can be used to describe the strong interac-

tion.

3.1 Running Coupling

The coupling for the strong interaction is the QCD gauge coupling, gs. We usually work in

terms of s defined as

s =g2s4

. (3.1)

Since the interactions are strong, we would expect s to be too large to perform reliable

calculations in perturbation theory. On the other hand the Feynman rules are only useful

within the context of perturbation theory.

This difficulty is resolved when we understand that coupling constants are not constant

at all. The electromagnetic fine structure constant, , has the value 1/137 only at energies

which are not large compared to the electron mass. At higher energies it is larger than this.For example, at LEP energies it takes a value close to 1 /129. In contrast to QED, it turns

out that in the non-abelian gauge theories of the Standard Model the weak and the strong

coupling decrease as the energy increases.

To see how this works within the context of QCD we note that when we perform higher

order perturbative calculations there are loop diagrams which have the effect of dressing

the couplings. For example, the one-loop diagrams which dress the coupling between a quark

and a gluon are:


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where

= + - -

are the diagrams needed to calculate the one-loop corrections to the gluon propagator.

These diagrams contain UV divergences and need to be renormalized, e.g. by subtracting atsome renormalization scale . This scale then appears inside a logarithm for the renormalized

quantities. This means that if the squared momenta of all the external particles coming into

the vertex are of order Q2, where Q , then the above diagrams give rise to a correctionwhich contains a logarithm of the ratio Q2/2:

2s 0 ln

Q2/2

. (3.2)

This correction is interpreted as the correction to the effective QCD coupling, s(Q2), at

momentum scale Q, i.e.

s(Q2) = s(

2) s(2)2 0 ln

Q2/2

+ . . . . (3.3)

The coefficient 0 is calculated to be

0 =11 Nc 2 nf

12 , (3.4)

where Nc is the number of colours (=3), nf is the number of active flavours, i.e. the number

of flavours whose mass threshold is below the momentum scale Q. Note that 0 is positive,

which means that the coefficient in front of the logarithm in eq. (3.3) is negative, so that the

effective coupling decreases as the momentum scale is increased.

A more precise analysis shows that the effective coupling obeys the differential equation

s(Q2)

ln(Q2)=

s(Q

2)

, (3.5)

where has the perturbative expansion

(s) = 0 2s 1 3s + O(4s) + . . . . (3.6)


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0

0.05

0.1

0.15

0.2

0.250.3

0.35

0.4

0.45

0.5

1 10 100

s(Q2)

Q2 (GeV)

Figure 3.1: The running of s(Q2) with taken to two loops.

In order to solve this differential equation we need a boundary value. Nowadays this is usually

taken to be the measured value of the coupling at scale of the Z boson mass, MZ = 91.19

GeV, which is measured to be

s(M2Z) = 0.118

0.002 . (3.7)

This is one of the free parameters of the Standard Model. 6

The running of s(Q2) is shown in figure 3.1. We can see that for momentum scales above

about 2 GeV the coupling is less than 0.3 so that one can hope to carry out reliable pertur-

bative calculations for QCD processes with energy scales larger than this.

Gauge invariance requires that the gauge coupling for the interaction between gluons must

be exactly the same as the gauge coupling for the interaction between quarks and gluons.

The -function could therefore have been calculated from the higher order corrections to thethree-gluon (or four-gluon) vertex and must yield the same result, despite the fact that it is

calculated from a completely different set of diagrams.

6Previously the solution to eq. (3.5) (to leading order) was written as s(Q2) = 4/0 ln(Q2/2QCD) and

the scale QCD was used as the standard parameter which sets the scale for the magnitude of the strong

coupling. This turns out to be rather inconvenient since it needs to be adjusted every time higher order

corrections are taken into consideration and the number of active flavours has to be specified. The detour

via QCD also introduces additional truncation errors and can complicate the error analysis.


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Exercise 3.1

Draw the Feynman diagrams needed for the calculation of the one-loop cor-

rection to the triple gluon coupling (dont forget the Faddeev-Popov ghost

loops).

Exercise 3.2

Solve equation (3.5) using to leading order only, and calculate the value of

s at a momentum scale of 10 GeV. Use the value at MZ given by eq. (3.7).

Calculate also the error in s at 10 GeV.

3.2 Quark (and Gluon) Confinement

This argument can be inverted to provide an answer to the question of why we have never seen

quarks or gluons in a laboratory. Asymptotic Freedom tells us that the effective coupling be-

tween quarks becomes weaker at shorter distances (equivalent to higher energies/momentum

scales). Conversely it implies that the effective coupling grows as we go to larger distances.

Therefore, the complicated system of gluon exchanges which leads to the binding of quarks

(and antiquarks) inside hadrons leads to a stronger and stronger binding as we attempt to

pull the quarks apart. This means that we can never isolate a quark (or a gluon) at large

distances since we require more and more energy to overcome the binding as the distance

between the quarks grows. Instead, when the energy contained in the string of bound glu-

ons and quarks becomes large enough, the colour-string breaks and more quarks are created,leaving more colourless hadrons, but no isolated, coloured quarks.

The upshot of this is that the only free particles which can be observed at macroscopic

distances from each other are colour singlets. This mechanism is known as quark confine-

ment. The details of how it works are not fully understood. Nevertheless the argument

presented here is suggestive of such confinement and at the level of non-perturbative field

theory, lattice calculations have confirmed that for non-abelian gauge theories the binding

energy does indeed grow as the distance between quarks increases.7

Thus we have two different pictures of the world of strong interactions: On one hand, at suf-

ficiently short distances, which can be probed at sufficiently large energies, we can consider

quarks and gluons (partons) interacting with each other. In this regime we can perform

calculations of the scattering cross sections between quarks and gluons (called the par-

tonic hard cross section) in perturbation theory because the running coupling is sufficiently

7Lattice QCD simulations have also succeeded in calculating the spectrum of many observed hadrons and

also hadronic matrix elements for certain processes from first principles, i.e. without using perturbative

expansions or phenomenological models.


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small. On the other hand, before we can make a direct comparison with what is observed

in accelerator experiments, we need to take into account the fact that the quarks and glu-

ons bind (hadronize) into colour singlet hadrons, and it is only these colour singlet states

that are observed directly. The mechanism for this hadronization is beyond the scope of

perturbation theory and not understood in detail. Nevertheless Monte Carlo programs havebeen developed which simulate the hadronization in such a way that the results of the short-

distance perturbative calculations at the level of quarks and gluons can be confronted with

experiments measuring hadrons in a successful way.

Thus, for example, if we wish to calculate the cross section for an electron-positron annihila-

tion into three jets (at high energies), we first calculate, in perturbation theory, the process

for electron plus positron to annihilate into a virtual photon (or Z boson) which then de-

cays into a quark and antiquark, and an emitted gluon. At leading order the two Feynman

diagrams for this process are:8

e+

e

q

q

g e+

e

q

q

g

However, before we can compare the results of this perturbative calculation with experi-

mental data on three jets of observed hadrons, we need to perform a convolution of this

calculated cross section with a Monte Carlo simulation that accounts for the way in which

the final state partons (quarks and gluons) bind with other quarks and gluons to produce

observed hadrons. It is only after such a convolution has been performed that one can get

a reliable comparison of the calculated observables (like cross sections or event shapes) with

data.

Likewise, if we want to calculate scattering processes including initial state hadrons we need

to account for the probability of finding a particular quark or gluon inside an initial hadron

with a given fraction of the initial hadrons momentum (these are called parton distribution

functions).

Exercise 3.3

Draw the (tree level) Feynman diagrams for the process e+e 4jets. Con-sider only one photon exchange plus the QCD contributions (do not include Z

boson exchange or W W production).

8The contraction of the one loop diagram (where a gluon connects the quark and antiquark) with the

e+e qq amplitude is of the same order s and has to be taken into account to get an infra-red finiteresult. However, it does not lead to a three-jet event (on the partonic level).


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3.3 -Parameter of QCD

There is one more gauge invariant term that can be written down in the QCD Lagrangian:

L = g2s

642

F

a

F

a

. (3.8)

Here is the totally antisymmetric tensor (in four dimensions). Since we should work

with the most general gauge invariant Lagrangian there is no reason to omit this term.

However, adding this term to the Lagrangian leads to a problem, called the strong CP

problem.

To understand the nature of the problem, we first convince ourselves that this term violates

CP. In QED we would have

FF = E B, (3.9)and for QCD we have a similar expression except that Ea and Ba carry a colour index

they are known as the chromoelectric and chromomagnetic fields. Under charge conjugation

both the electric and magnetic field change sign. But under parity the electric field, which

is a proper vector, changes sign, whereas the magnetic field, which is a polar vector, does

not change sign. Thus we see that the term E B is odd under CP.For this reason, the parameter in front of this term must be exceedingly small in order not

to give rise to strong interaction contributions to CP violating quantities such as the electric

dipole moment of the neutron. The current experimental limits on this dipole moment tell

us that < 1010. Thus we are tempted to think that is zero. Nevertheless, strictly

speaking is a free parameter of QCD, and is sometimes considered to be the nineteenth

free parameter of the Standard Model.

Of course we simply could set to zero (or a very small number) and be happy with it.9

However, whenever a free parameter is zero or extremely small, we would like to understand

the reason. The fact that we do not know why this term is absent (or so small) is the strong

CP problem.

There are several possible solutions to the strong CP problem that offer explanations as

to why this term is absent (or small). One possible solution is through imposing an ad-

ditional symmetry, leading to the postulation of a new, hypothetical, weakly interacting

particle, called the (Peccei-Quinn) axion. Unfortunately none of these solutions have been

confirmed yet and the problem is still unresolved.

Another question is why is this not a problem in QED? In fact a term like eq. (3.8) can also

9To be precise, setting 0 in the Lagrangian would not be enough, as = 0 can also be generatedthrough higher order electroweak radiative corrections, requiring a fine-tuning beyond 0.


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be written down in QED. A thorough discussion of this point is beyond the scope of this

lecture. Suffice to say that this term can be written (in QED and QCD) as a total divergence,

so it seems that it can be eliminated from the Lagrangian altogether. However, in QCD (but

not in QED) there are non-perturbative effects from the non-trivial topological structure of

the vacuum (somewhat related to so called instantons you probably have heard about)which prevent us from neglecting the -term.

3.4 Summary

Quarks transform as a triplet representation of colour SU(3) (each quark can have oneof three colours).

The eight gauge bosons of QCD are the gluons which are the carriers that mediate the

strong interaction.

The coupling of quarks to gluons (and gluons to each other) decreases as the energyscale increases. Therefore, at high energies one can perform reliable perturbative cal-

culations for strongly interacting processes.

As the distance between quarks increases the binding increases, such that it is impos-sible to isolate individual quarks or gluons. The only observable particles are colour

singlet hadrons. Perturbative calculations performed at the quark and gluon level must

be supplemented by accounting for the recombination of final state quarks and gluonsinto observed hadrons as well as the probability of finding these quarks and gluons

inside the initial state hadrons (if applicable).

QCD admits a gauge invariant strong CP violating term with a coefficient . Thisparameter is known to be very small from limits on CP violating phenomena such as

the electric dipole moment of the neutron.


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4 Spontaneous Symmetry Breaking

We have seen that in an unbroken gauge theory the gauge bosons must be massless. This is

exactly what we want for QED (massless photon) and QCD (massless gluons). However, if we

wish to extend the ideas of describing interactions by a gauge theory to the weak interactions,the symmetry must somehow be broken since the carriers of the weak interactions (W and

Z bosons) are massive (weak interactions are very short range). We could simply break the

symmetry by hand by adding a mass term for the gauge bosons, which we know violates the

gauge symmetry. However, this would destroy renormalizability of our theory.

Renormalizable theories are preferred because they are more predictive. As discussed in

the Field Theory and QED lectures, there are divergent results (infinities) in QED and

QCD, and these are said to be renormalizable theories. So what could be worse about

a non-renormalizable theory? The critical issue is the number of divergences: few in arenormalizable theory, and infinite in the non-renormalizable case. Associated to every

divergence is a parameter that must be extracted from data, so renormalizable theories can

make testable predictions once a few parameters are measured. For instance, in QCD, the

coupling gs has a divergence. But once s is measured in one process, the theory can be

tested in other processes.10

In this chapter we will discuss a way to give masses to the W and Z, called spontaneous

symmetry breaking, which maintains the renormalizability of the theory. In this scenario

the Lagrangian maintains its symmetry under a set of local gauge transformations. On theother hand, the lowest energy state, which we interpret as the vacuum (or ground state),

is not a singlet of the gauge symmetry. There is an infinite number of states each with the

same ground-state energy and nature chooses one of these states as the true vacuum.

4.1 Massive Gauge Bosons and Renormalizability

In this subsection we will convince ourselves that simply adding by hand a mass term for

the gauge bosons will destroy the renormalizability of the theory. It will not be a rigorousargument, but will illustrate the difference between introducing mass terms for the gauge

bosons in a brute force way and introducing them via spontaneous symmetry breaking.

Higher order (loop) corrections generate ultraviolet divergences. In a renormalizable theory,

10It should be noted that effective field theories, though formally not renormalizable, can nevertheless be

very valuable as they often allow for a simplified description of a more complete or fundamental theory in

a resticted energy range. Popular examples are Chiral Perturbation Theory, Heavy Quark Effective Theory

and Non-Relativistic QCD.


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these divergences can be absorbed into the parameters of the theory we started with, and

in this way can be hidden. As we go to higher orders we need to absorb more and more

terms into these parameters, but there are only as many divergent quantities as there are

parameters. So, for instance, in QED the Lagrangian we start with contains the fermion

field, the gauge boson field, and interactions whose strength is controlled by e and m. Beinga renormalizable theory, all divergences of diagrams can be absorbed into these quantities

(irrespective of the number of loops or legs), and once e and m are measured, all other

observables (cross sections, g 2, etc.) can be predicted.In order to ensure that this programme can be carried out there have to be restrictions on

the allowed interaction terms. Furthermore all the propagators have to decrease like 1/p2

as the momentum p . Note that this is how the massless gauge-boson propagatoreq. (1.24) behaves. If these conditions are not fulfilled, then the theory generates more and

more divergent terms as one calculates to higher orders, and it is not possible to absorbthese divergences into the parameters of the theory. Such theories are said to be non-

renormalizable.

Now we can convince ourselves that simply adding a mass term M2 AA to the Lagrangian

given in eq. (2.21) will lead to a non-renormalizable theory. To start with we note that

such a term will modify the propagator. Collecting all terms bilinear in the gauge fields in

momentum space we get (in Feynman gauge)

1

2A g(p2 M2) + ppA. (4.1)

We have to invert this operator to get the propagator which now takes the form

i

p2 M2g+ p

p

M2

. (4.2)

Note that this propagator, eq. (4.2), has a much worse ultraviolet behavior in that it goes

to a constant for p . Thus, it is clear that the ultraviolet properties of a theory witha propagator as given in eq. (4.2) are worse than for a theory with a propagator as given

in eq. (1.24). According to our discussion at the beginning of this subsection we conclude

that without the explicit mass term M2 AA the theory is renormalizable, whereas with

this term it is not. In fact, it is precisely the gauge symmetry that ensures renormalizability.

Breaking this symmetry results in the loss of renormalizability.

The aim of spontaneous symmetry breaking is to break the gauge symmetry in a more subtle

way, such that we can still give the gauge bosons a mass but retain renormalizability.


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4.2 Spontaneous Symmetry Breaking

Spontaneous symmetry breaking is a phenomenon that is by far not restricted to gauge

symmetries. It is a subtle way to break a symmetry by still requiring that the Lagrangian

remains invariant under the symmetry transformation. However, the ground state of thesymmetry is not invariant, i.e. not a singlet under a symmetry transformation.

In order to illustrate the idea of spontaneous symmetry breaking, consider a pen that is

completely symmetric with respect to rotations around its axis. If we balance this pen on

its tip on a table, and start to press on it with a force precisely along the axis we have a

perfectly symmetric situation. This corresponds to a Lagrangian which is symmetric (under

rotations around the axis of the pen in this case). However, if we increase the force, at some

point the pen will bend (and eventually break). The question then is in which direction will

it bend. Of course we do not know, since all directions are equal. But the pen will pick

one and by doing so it will break the rotational symmetry. This is spontaneous symmetry

breaking.

A better example can be given by looking at a point mass in a potential

V(r) = 2r r + (r r)2. (4.3)

This potential is symmetric under rotations and we assume > 0 (otherwise there would

be no stable ground state). For 2 > 0 the potential has a minimum at r = 0, thus the

point mass will simply fall to this point. The situation is more interesting if 2 < 0. For

two dimensions the potential is shown in Fig. 4.1. If the point mass sits at r = 0 the

system is not in the ground state but the situation is completely symmetric. In order to

reach the ground state, the symmetry has to be broken, i.e. if the point mass wants to roll

down, it has to decide in which direction. Any direction is equally good, but one has to be

picked. This is exactly what spontaneous symmetry breaking means. The Lagrangian (here

the potential) is symmetric (here under rotations around the z-axis), but the ground state

(here the position of the point mass once it rolled down) is not. Let us formulate this in

a slightly more mathematical way for gauge symmetries. We denote the ground state by

|0. A spontaneously broken gauge theory is a theory whose Lagrangian is invariant undergauge transformations, which is exactly what we have done in chapters 1 and 2. The new

feature in a spontaneously broken theory is that the ground state is not invariant under

gauge transformations. This means

eiaTa |0 = |0 (4.4)

which entails

Ta |0 = 0 for some a. (4.5)


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y

V(r)

x

Figure 4.1: A potential that leads to spontaneous symmetry breaking.

Eq. (4.5) follows from eq. (4.4) upon expansion in a. Thus, the theory is spontaneously

broken if there exists at least one generator that does not annihilate the vacuum.

In the next section we will explore the concept of spontaneous symmetry breaking in the

context of gauge symmetries in more detail, and we will see that, indeed, this way of breaking

the gauge symmetry has all the desired features.

4.3 The Abelian Higgs Model

For simplicity, we will start by spontaneously breaking the U(1) gauge symmetry in a theory

of one complex scalar field. In the Standard Model, it will be a non-abelian gauge theory

that is spontaneously broken, but all the important ideas can simply be translated from the

U(1) case considered here.

The Lagrangian density for a gauged complex scalar field, with a mass term and a quartic

self-interaction, may be written as

L = (D) D 14

FF V(), (4.6)

where the potential V(), is given by

V() = 2 + ||2 , (4.7)


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and the covariant derivative D and the field-strength tensor F are given in eqs. (1.15) and

(1.12) respectively. This Lagrangian is invariant under U(1) gauge transformations

ei(x). (4.8)

Provided 2 is positive this potential has a minimum at = 0. We call the = 0 state

the vacuum and expand in terms of creation and annihilation operators that populate the

higher energy states. In terms of a quantum field theory, where is an operator, the precise

statement is that the operator has zero vacuum expectation value, i.e. 0||0 = 0.Now suppose we reverse the sign of 2, so that the potential becomes

V() = 2 + ||2 , (4.9)

with 2 > 0. We see that this potential no longer has a minimum at = 0, but a (local)

maximum. The minimum occurs at

= ei

2

2 ei v

2, (4.10)

where can take any value from 0 to 2. There is an infinite number of states each with

the same lowest energy, i.e. we have a degenerate vacuum. The symmetry breaking occurs

in the choice made for the value of which represents the true vacuum. For convenience we

shall choose = 0 to be our vacuum. Such a choice constitutes a spontaneous breaking of

the U(1) invariance, since a U(1) transformation takes us to a different lowest energy state.In other words the vacuum breaks U(1) invariance. In quantum field theory we say that the

field has a non-zero vacuum expectation value

= v2

. (4.11)

But this means that there are excitations with zero energy, that take us from the vacuum to

one of the other states with the same energy. The only particles which can have zero energy

are massless particles (with zero momentum). We therefore expect a massless particle in

such a theory.

To see that we do indeed get a massless particle, let us expand around its vacuum expec-

tation value,

=ei/v

2

+ H

1

2

+ H+ i

. (4.12)

The fields H and have zero vacuum expectation values and it is these fields that are

expanded in terms of creation and annihilation operators of the particles that populate the

excited states. Of course, it is the H-field that corresponds to the Higgs field.


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We now want to write the Lagrangian in terms of the H and fields. In order to get the

potential we insert eq. (4.12) into eq. (4.9) and find

V = 2H2 +

H3 + 2H

+

4 H4 + 4 + 2H2 2

+

4

4. (4.13)

Note that in eq. (4.13) there is a mass term for the H-field, 2H2 MH/2H2, where wehave defined11

MH =

2. (4.14)

However, there is no mass term for the field . Thus is a field for a massless particle called

the Goldstone boson. We will look at this issue in a more general way in section 4.4. Next

let us consider the kinetic term. We plug eq. (4.12) into (D)D and get

(D) D =

1

2

HH+

1

2

+

1

2

g2v2AA +

1

2

g2AA(H2 + 2)

gA (H H) + gvA + g2vAAH. (4.15)

There are several important features in eq. (4.15). Firstly, the gauge boson has acquired a

mass term 1/2g2v2AA 1/2M2AAA, where we have defined

MA = gv. (4.16)

Secondly, there is a coupling of the gauge field to the H-field,

g2

vAA

H = gMAAA

H. (4.17)

It is important to remember that this coupling is proportional to the mass of the gauge

boson. Finally, there is also the bilinear term g v A, which after integrating by parts

(for the action S) may be written as MA A. This mixes the Goldstone boson, , withthe longitudinal component of the gauge boson, with strength MA (when the gauge-boson

field A is separated into its transverse and longitudinal components, A = AL + A

T ,

where AT = 0). Later on, we will use the gauge freedom to get rid of this mixing term.

4.4 Goldstone Bosons

In the previous subsection we have seen that there is a massless boson, called the Goldstone

boson, associated with the flat direction in the potential. Goldstones theorem describes the

appearance of massless bosons when a global (not gauge) symmetry is spontaneously broken.

11Note that for a real field representing a particle of mass m the mass term is 12m22, whereas for a

complex field the mass term is m2.


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Suppose we have a theory whose Lagrangian is invariant under a symmetry group G withN generators Ta and the symmetry group of the vacuum forms a subgroup H of G, withm generators. This means that the vacuum state is still invariant under transformations

generated by the m generators ofH, but not the remaining N m generators of the originalsymmetry group G. Thus we have

Ta|0 = 0 a = 1 . . . m ,Ta|0 = 0 a = m + 1 . . . N . (4.18)

Goldstones theorem states that there will be N

m massless particles (one for each broken

generator of the group). The case considered in this section is special in that there is only

one generator of the symmetry group (i.e. N = 1) which is broken by the vacuum. Thus,

there is no generator that leaves the vacuum invariant (i.e. m = 0) and we get N m = 1Goldstone boson.

Like all good general theorems, Goldstones theorem has a loophole, which arises when one

considers a gauge theory, i.e. when one allows the original symmetry transformations to

be local. In a spontaneously broken gauge theory, the choice of which vacuum is the true

vacuum is equivalent to choosing a gauge, which is necessary in order to be able to quantize

the theory. What this means is that the Goldstone bosons, which can, in principle, transform

the vacuum into any of the states degenerate with the vacuum, now affect transitions into

states which are not consistent with the original gauge choice. This means that the Goldstone

bosons are unphysical and are often called Goldstone ghosts.

On the other hand the quantum degrees of freedom associated with the Goldstone bosons

are certainly there ab initio (before a choice of gauge is made). What happens to them? A

massless vector boson has only two degrees of freedom (the two directions of polarization

of a photon), whereas a massive vector (spin-one) particle has three possible values for

the helicity of the particle. In a spontaneously broken gauge theory, the Goldstone bosonassociated with each broken generator provides the third degree of freedom for the gauge

bosons. This means that the gauge bosons become massive. The Goldstone boson is said to

be eaten by the gauge boson. This is related to the mixing term between AL and of the

previous subsection. Thus, in our abelian model, the two degrees of freedom of the complex

field turn out to be the Higgs field and the longitudinal component of the (now massive)

gauge boson. There is no physical, massless particle associated with the degree of freedom

present in .


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4.5 The Unitary Gauge

As mentioned above, we want to use the gauge freedom to choose a gauge such that there are

no mixing terms between the longitudinal component of the gauge field and the Goldstone

boson. Recall =

12

(v + H) ei/v =1

2

+ H+ i + . . .

, (4.19)

where the dots stand for nonlinear terms in . Next we make a gauge transformation (see

eq. (1.2))

= ei/v . (4.20)In other words, we fix the gauge such that the imaginary part of vanishes. Under the

gauge transformation eq. (4.20) the gauge field transforms according to (see eq. (1.11))

A A

= A +

1

gv []. (4.21)

It is in fact the superposition of A and which make up the physical field. Note that

the change from A to A made in eq. (4.21) affects only the longitudinal component. If

we now express the Lagrangian in terms of and A there will be no mixing term. Even

better, the field vanishes altogether! This can easily be seen by noting that under a gauge

transformation the covariant derivative D transforms in the same way as , thus

D (D) = ei/vD = ei/v 12

H+ igA

(v + H)

, (4.22)

and (D)(D) is independent of . Performing the algebra (and dropping the for the

A-field) we get the Lagrangian in the unitary gauge

L = 12

HH+

M2A2

AA 1

4FF

M2H

2H2

+ gMAAAH+

g2

2AA

H2 4

H4

2MHH

3, (4.23)

with MA and MH as defined in eqs. (4.16) and (4.14), respectively. All the terms quadratic

in A may be written (in momentum space) as

A(p)gp2 + pp+ gM2A

A(p). (4.24)

The gauge boson propagator is the inverse of the coefficient of A(p)A(p), which is

i

g ppM2A

1

(p2 M2A). (4.25)

This is the usual expression for the propagator of a massive spin-one particle, eq. (4.2).

The only other remaining particle is the scalar, H, with mass mH =

2 , which is the


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Higgs boson. This is a physical particle, which interacts with the gauge boson and also has

cubic and quartic self-interactions. The Lagrangian given in eq. (4.23) leads to the following

vertices and Feynman rules:

2 i e2g

2 i eMAg

6 i

6 i mH

2

The advantage of the unitary gauge is that no unphysical particles appear, i.e. the -field

has completely disappeared. The disadvantage is that the propagator of the gauge field,

eq. (4.25), behaves as p0

for p . As discussed in section 4.1 this seems to indicatethat the theory is non-renormalizable. It seems that we have not gained anything at all

by breaking the theory spontaneously rather than by simply adding a mass term by hand.

Fortunately this is not true. In order to see that the theory is still renormalizable, in spite

of eq. (4.25), it is very useful to consider a different type of gauges, namely the R gauges

discussed in the next subsection.

4.6 R Gauges (Feynman Gauge)

The class of R gauges is a more conventional way to fix the gauge. Recall that in QED we

fixed the gauge by adding a term, eq. (1.21), in the Lagrangian. This is exactly what we do

here. The gauge fixing term we are adding to the Lagrangian density eq. (4.6) is

LR 12(1 ) (A

(1 )MA)2

= 12(1 )A

A+ MAA

1 2

M2A2. (4.26)


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Again, the special value = 0 corresponds to the Feynman gauge. The second term in

eq. (4.26) cancels precisely the mixing term in eq. (4.15). Thus, we have achieved our goal.

Note however, that in this case, contrary to the unitary gauge, the unphysical -field does

not disappear. The first term in eq. (4.26) is bilinear in the gauge field, thus it contributes

to the gauge-boson propagator. The terms bilinear in the A-field are

12

A(p)g(p2 M2a ) + pp

pp1

A(p) (4.27)

which leads to the gauge boson propagator

i(p2 M2A)

g pp

p2 (1 )M2A

. (4.28)

In the Feynman gauge, the propagator becomes particularly simple. The crucial feature of

eq. (4.28), however, is that this propagator behaves as p2 for p . Thus, this classof gauges is manifestly renormalizable. There is, however, a price to pay: The Goldstone

boson is still present. It has acquired a mass, MA, from the gauge fixing term, and it has

interactions with the gauge boson, with the Higgs scalar and with itself. Furthermore, for the

purposes of higher order corrections in non-Abelain theories, we need to introduce Faddeev-

Popov ghosts which interact with the gauge bosons, the Higgs scalar and the Goldstone

bosons.

Let us stress that there is no contradiction at all between the apparent non-renormalizability

of the theory in the unitary gauge and the manifest renormalizability in the R gauge. Sincephysical quantities are gauge invariant, any physical quantity can be calculated in a gauge

where renormalizability is manifest. As mentioned above, the price we pay for this is that

there are more particles and many more interactions, leading to a plethora of Feynman

diagrams. We therefore only work in such gauges if we want to compute higher order

corrections. For the rest of these lectures we shall confine ourselves to tree-level calculations

and work solely in the unitary gauge.

Nevertheless, one cannot over-stress the fact that it is only when the gauge bosons ac-

quire masses through the Higgs mechanism that we have a renormalizable theory. It is thismechanism that makes it possible to write down a consistent Quantum Field Theory which

describes the weak interactions.

4.7 Summary

In the case of a gauge theory the Goldstone bosons provide the longitudinal componentof the gauge bosons, which therefore acquire a mass. The mass is proportional to the


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magnitude of the vacuum expectation value and the gauge coupling constant. The

Goldstone bosons themselves are unphysical.

It is possible to work in the unitary gauge where the Goldstone boson fields are set tozero.

When gauge bosons acquire masses by this (Higgs) mechanism, renormalizability ismaintained. This can be seen explicitly if one works in a R gauge, in which the gauge

boson propagator decreases like 1/p2 as p . This is a necessary condition forrenormalizability. If one does work in such a gauge, however, one needs to work with

Goldstone boson fields, even though the Goldstone bosons are unphysical. The number

of interactions and the number of Feynman graphs required for the calculation of some

processes is then greatly increased.


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5 The Standard Model with one Family

To write down the Lagrangian of a theory, one first needs to choose the symmetries (gauge

and global) and the particle content, and then write down every allowed renormalizable

interaction. In this section we shall use this recipe to construct the Standard Model withone family. The Lagrangian should contain pieces

L(SM,1) = Lgauge bosons + Lfermion masses + LfermionKT + LHiggs. (5.1)

The terms are written out in eqns. (5.15), (5.29), (5.30) and (5.55).

5.1 Left- and Right- Handed Fermions

The weak interactions are known to violate parity. Parity non-invariant interactions forfermions can be constructed by giving different interactions to the left-handed and right-

handed components defined in eq. (5.4). Thus, in writing down the Standard Model, we

will treat the left-handed and right-handed parts separately.

A Dirac field, , representing a fermion, can be expressed as the sum of a left-handed part,

L, and a right-handed part, R,

= L + R, (5.2)

where

L = PL with PL =(1 5)

2, (5.3)

R = PR with PR =(1 + 5)

2. (5.4)

PL and PR are projection operators, i.e.

PL PL = PL, PR PR = PR and PL PR = 0 = PR PL. (5.5)

They project out the left-handed (negative) and right-handed (positive) chirality states ofthe fermion, respectively. This is the definition of chirality, which is a property of fermion

fields, but not a physical observable.

The kinetic term of the Dirac Lagrangian and the interaction term of a fermion with a vector

field can also be written as a sum of two terms, each involving only one chirality

= L L + R

R, (5.6)

A = L AL + R

AR. (5.7)


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On the other hand, a mass term mixes the two chiralities:

m = mL R + mR L. (5.8)

Exercise 5.1

Use (5)2 = 1 to verify eq. (5.5) and = 0, 5 = 5 as well as 5 =

5 to verify eq. (5.7).

In the limit where the fermions are massless (or sufficiently relativistic), chirality becomes

helicity, which is the projection of the spin on the direction of motion and which is a physical

observable. Thus, if the fermions are massless, we can treat the left-handed and right-handed

chiralities as separate particles of conserved helicity. We can understand this physically from

the following simple consideration. If a fermion is massive and is moving in the positive z

direction, along which its spin is having a positive component so that the helicity is positive

in this frame, one can always boost into a frame in which the fermion is moving in the

negative z direction, but with this spin component unchanged. In the new frame the helicity

will hence be negative. On the other hand, if the particle is massless and travels with the

speed of light, no such boost is possible, and in that case helicity/chirality is a good quantum

number.

Exercise 5.2For a massless spinor

u(p) =1E

E p

,where is a two-component spinor, show that

(1 5)u(p)

are eigenstates of p/E with eigenvalues 1, respectively. Take

5 = 0 1

1 0

,and in 4 4 matrix notation p means p 0

0 p

.


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5.2 Symmetries and Particle Content

We have made all the preparations to write down a gauge invariant Lagrangian. We now

only have to pick the gauge group and the matter content of the theory. It should be noticed

that there are no theoretical reasons to pick a certain group or certain matter content. Tomatch experimental observations we pick the gauge group for the Standard Model to be

U(1)Y SU(2) SU(3). (5.9)

To indicate that the abelian U(1) group is not the gauge group of QED but of hypercharge

a subscript Y has been added. The corresponding coupling and gauge boson is denoted by

g and B respectively.

The SU(2) group has three generators (Ta = a/2), the coupling is denoted by

thomas teubner - standard model (2008)

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